In this chapter we give some indication of how Thurston originally discovered the Nielsen–Thurston classification theorem. We begin with a concrete, accessible example that illustrates much of the general theory. We then provide a sketch of how that general theory works. Our goal is not to give a formal treatment as per the rest of the text. Rather, we hope to convey to the reader part of the beautiful circle of ideas surrounding the Nielsen–Thurston classification, including Teichmüller’s theorems, Markov partitions, train tracks, foliations, laminations, and more.